It sounds like your son is having trouble with being disappointed with not performing perfectly well, not winning, not getting praised and with being criticized. He's having trouble with the concept of being a team player and fair play competition.
What he needs is more practice in being gracious when he's feeling disappointment and in controlling his anger. He needs encouragement and praise for his positive behavior. He should have goals to work toward and good examples in the home of how to handle disappointment and anger.
Remember, though, that he's only five years old? He can't possibly automatically know how to control his emotions and behavior at this age. He needs help. It doesn't mean he has anger issues or that he's "an angry child." He's not bad either. Like other children his age he has impulse control problems and they're tied to his emotions. He just needs a lot of help with it.
I would put some effort into having him start by apologizing to people that he's pushed/injured and shouted at.